Exterior boundary‐value problems are considered for the class of cylinders defined byr=r(ϑ), with 1/rexpressible as a finite Fourier series, so thatr(ϑ) is meromorphic in the complex ϑ plane. Upon transforming the extended boundary condition integrals into the complex plane, simple and apparently exact asymptotic behavior is determined for the surface field expansion coefficientsbm; e.g., for a charged cylinder with no applied field one hasbm∼2∑exp(imϑp), form≫1, where the sum is over poles ofr(ϑ) in the upper half‐plane. These results are shown to have significant impact on convergence for solutions to the infinite systems of equations involved. We consider potential theory for Dirichlet and Neumann boundary conditions, and acoustic scattering in the Rayleigh limit.